What was done
The authors discuss a number of complexities associated with the assessment of global sea level change, review the current state of our knowledge about this important subject, and give their prognosis for the future.

What was learned
Mean global sea level (GSL), according to the authors, was relatively stable for the past few millennia, but it "abruptly began to rise near the mid-19th century."  In this regard, however, they note that no studies "have detected any significant acceleration of GSL rise during the 20th century."

The authors' best estimate of the mean rate of GSL rise over the past century is, as they put it, "closer to 2 mm/y than 1 mm/y."  With respect to reasonably well understood sources of water that likely contribute to this rise, they say that 0.6 mm/y likely comes from thermal expansion of the oceans and that 0.3 mm/y likely comes from the melting of small ice sheets and glaciers, leaving about 1 mm/y to be explained (their "puzzle"), if their best estimate of GSL rise is correct.

How will the puzzle be solved?  The authors describe new satellite programs designed to provide the answer; but they say that "several years of data and further efforts at interpretation will doubtless be required."

What it means
At the current time, we know two things about mean global sea level: (1) after a several-millennia hiatus, global sea level began to rise about 1850, and (2) the rise since that time, to the best of our knowledge, has been at a constant rate.  We also know that the air's CO2 content began to rise at about the same time, i.e., 1850.  However, its rate of rise has been anything but constant.  Over the first half of this period (1850 to 1925), for example, the atmosphere gained about 20 ppm of CO2, whereas over the second half (1925-2000) it gained about 65 ppm.  Hence, it would appear to be a good bet that the historical rise in mean global sea level has not been driven by the concomitant rise in the air's CO2 content.